Coaxial scattering of Euler-equation translating V-states via contour dynamics.

*(English)*Zbl 0581.76025The robustness of localized states which transport energy and mass is assessed by a numerical study of the Euler equation in two space dimensions. The localized states are the translating ”V-states” discovered by G. S. Deem and the author [Phys. Rev. Lett. 40, 859- 862 (1978) and in: K. Lonngren and A. Scott (eds.), Solitons in action (1978; Zbl 0494.76022) on pp. 277-293]. These piecewise- constant dipolar (i.e., oppositely-signed \(\pm\) or \(\mp)\) vorticity regions are steady translating solutions of the Euler equations. A new adaptive contour-dynamical algorithm with curvature-controlled node insertion and removal is used. The evolution of one V-state, subject to a symmetric-plus-asymmetric perturbation is examined and stable (i.e, nondivergent) fluctuations are observed. For scattering interactions, coaxial head-on (or \(\pm\) on \(\mp)\) and head-tail (or \(\pm\) on \(\pm)\) arrangements are studied. The temporal variation of contour curvature and perimeter after V-states separate indicate that internal degrees of freedom have been excited. For weak interactions we observe phase shifts and the near recurrence to initial states. When two similar, equal- circulation but unequal-area V-states have a head-on interaction a new asymmetric state is created by contour ”exchange”. There is strong evidence that this is near to a V-state. For strong interactions we observe phase shifts, ”breaking” (filament formation) and, for head-tail interactions, merger of like-signed vorticity regions.

##### MSC:

76B47 | Vortex flows for incompressible inviscid fluids |

##### Keywords:

dipolar vorticity region; coaxial scattering; robustness of localized states; Euler equation in two space dimensions; steady translating solutions; contour-dynamical algorithm; curvature-controlled node insertion; evolution of one V-state; symmetric-plus-asymmetric perturbation; scattering interactions; phase shifts; recurrence; filament formation
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\textit{E. A. Overman} and \textit{N. J. Zabusky}, J. Fluid Mech. 125, 187--202 (1982; Zbl 0581.76025)

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